Aspects of functional data inference and its applications
Lee, Jong Soo
Cox, Dennis D.
Doctor of Philosophy
We consider selected topics in estimation and testing of functional data. In many applications of functional data analysis, we aim to compare the sample functional data from two or more populations. However, the raw functional data often contains noise, some of which can be huge outliers. Hence, we must first perform the smoothing and estimation of functional data, but the existing methods for robust smoothing parameter selection are unsatisfactory. We present an efficient way to compute a smoothing parameter which can be generally applied to most robust smoothers. Then, we propose a procedure for testing pointwise difference of functional data in the two-sample framework. Our proposed method is a generalization of Hotelling's T2 test, and we utilize an adaptive truncation technique of Fan and Lin (1998) for dimension reduction and development of the test statistic. We show that our method performs well when compared with the existing testing procedures. Furthermore, we propose a method to detect the significantly different regions between curves. Once we determine that the samples curves from the two or more populations are significantly different overall, we want to look at the local regions of the curves and see where the differences occur. We present a modification of the multiple testing procedure of Westfall and Young (1993) for this testing method. Finally, we apply our proposed methods to the data from the fluorescence spectroscopic device. The fluorescence spectroscopic device is a medical device designed for early detection of cervical cancer, and the output from the device is a functional data, which makes the analysis challenging. The problems posed by this application have motivated the development of the methodologies in the present work, and we demonstrate that our methods work well in this application.